Is there somewhere to park on the US side if you want to walk across for the day?

Yes before exiting 6 dollars a day and your car is save walking distance is about a block

Yes. But somehow the stores there, particularly the only grocery store (T.C. Worthy) charges to park your car there, although I don't know how they would get a tow truck from the city to come tow your vehicle quick enough. If you plan on staying over night in MX, you might want to pay the $5 just to be safe, to park at their location. There is additional parking around there as well. Some charge and there are spots where you can get lucky and park w/o paying at all.